## Algebraic & Geometric Topology

### A rational splitting of a based mapping space

#### Abstract

Let $ℱ∗(X,Y)$ be the space of base-point-preserving maps from a connected finite CW complex $X$ to a connected space $Y$. Consider a CW complex of the form $X∪αek+1$ and a space $Y$ whose connectivity exceeds the dimension of the adjunction space. Using a Quillen–Sullivan mixed type model for a based mapping space, we prove that, if the bracket length of the attaching map $α:Sk→X$ is greater than the Whitehead length $WL(Y)$ of $Y$, then $ℱ∗(X∪αek+1,Y)$ has the rational homotopy type of the product space $ℱ∗(X,Y)×Ωk+1Y$. This result yields that if the bracket lengths of all the attaching maps constructing a finite CW complex $X$ are greater than $WL(Y)$ and the connectivity of $Y$ is greater than or equal to $dimX$, then the mapping space $ℱ∗(X,Y)$ can be decomposed rationally as the product of iterated loop spaces.

#### Article information

Source
Algebr. Geom. Topol., Volume 6, Number 1 (2006), 309-327.

Dates
Revised: 14 February 2006
Accepted: 14 February 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796514

Digital Object Identifier
doi:10.2140/agt.2006.6.309

Mathematical Reviews number (MathSciNet)
MR2220679

Zentralblatt MATH identifier
1097.55010

Subjects
Primary: 55P62: Rational homotopy theory

#### Citation

Kuribayashi, Katsuhiko; Yamaguchi, Toshihiro. A rational splitting of a based mapping space. Algebr. Geom. Topol. 6 (2006), no. 1, 309--327. doi:10.2140/agt.2006.6.309. https://projecteuclid.org/euclid.agt/1513796514

#### References

• I Berstein, T Ganea, Homotopical nilpotency, Illinois J. Math. 5 (1961) 99–130
• E H Brown, Jr, R H Szczarba, On the rational homotopy type of function spaces, Trans. Amer. Math. Soc. 349 (1997) 4931–4951
• Y Félix, S Halperin, J-C Thomas, Differential graded algebras in topology, from: “Handbook of algebraic topology”, North-Holland, Amsterdam (1995) 829–865
• Y Félix, S Halperin, J-C Thomas, Rational homotopy theory, Graduate Texts in Mathematics 205, Springer, New York (2001)
• J B Friedlander, S Halperin, An arithmetic characterization of the rational homotopy groups of certain spaces, Invent. Math. 53 (1979) 117–133
• P Hilton, G Mislin, J Roitberg, Localization of nilpotent groups and spaces, North-Holland Publishing Co., Amsterdam (1975)
• S Kaji, On the nilpotency of rational $H$-spaces, J. Math. Soc. Japan 57 (2005) 1153–1165
• Y Kotani, Note on the rational cohomology of the function space of based maps, Homology Homotopy Appl. 6 (2004) 341–350
• K Kuribayashi, A rational model for the evaluation map, to appear in Georgian Mathematical Journal 13 (2006)